Testing Critical Slowing Down as a Bifurcation Indicator in a Low-dissipation Dynamical System
M. Marconi, C. Metayer, A. Acquaviva, J.M. Boyer, A. Gomel, T. Quiniou, C. Masoller, M. Giudici, J.R. Tredicce
TTesting Critical Slowing Down as a Bifurcation Indicatorin a Low-dissipation Dynamical System
M. Marconi, C. M´etayer, A. Acquaviva, J.M. Boyer, A. Gomel, T. Quiniou, C. Masoller, M. Giudici, and J.R. Tredicce
2, 3 Universit´e Cˆote d´ Azur, Institut de Physique de Nice, CNRS - UMR 7010, Sophia Antipolis, France Universit´e de la Nouvelle Cal´edonie, ISEA, BP R4 - 98851 Noum´ea Cedex, Nouvelle Cal´edonie Universidad de Buenos Aires, Departamento de Fisica,Intendente Guiraldes 2160, CABA, Buenos Aires, Argentina Departamento de Fisica, Universitat Politecnica de Catalunya, St Nebridi 22, Barcelona 08222, Spain. ∗ (Dated: August 25, 2020)We study a two-dimensional low-dissipation dynamical system with a control parameter that isswept linearly in time across a transcritical bifurcation. We investigate the relaxation time of aperturbation applied to a variable of the system and we show that critical slowing down may occurat a parameter value well above the bifurcation point. We test experimentally the occurrence ofcritical slowing down by applying a perturbation to the accessible control parameter and we findthat this perturbation leaves the system behavior unaltered, thus providing no useful informationon the occurrence of critical slowing down. The theoretical analysis reveals the reasons why thesetests fail in predicting an incoming bifurcation. PACS numbers: 42.55.-f, 03.65.Sq, 05.70.Fh
There has always been a special interest in trying topredict transitions, crisis, and catastrophic events. To-day, a huge amount of research is devoted to determinegood indicators applicable on time series obtained fromreal systems that may anticipate a change in its behav-ior or, in the language of dynamical systems, a bifur-cation [1–3]. This is particularly relevant in disciplineslike medicine, biology, atmospheric science, ecology, so-ciology, economy, where these predictions may avoid adisaster or, at least, they may be useful to prepare thesystem to a behavioral change. For example, it has beenconjectured that the advent of an epilepsy attack is theresult of a phase transition [4, 5], that climate on earth isactually very close to a tipping point [6], that extremelyintense pulses in lasers may result from a bifurcation of achaotic attractor [7, 8] and that evolutive specializationin ecology [9, 10] is also the consequence of a bifurcation.We may say that any behavioral change in a real systemis connected to the existence of a bifurcation in the cor-responding dynamical system and that the prediction ofthese changes depends on the possibility of establishingreliable indicators alerting of the incoming bifurcation.A well-established indicator which follows from the def-inition of bifurcation is known as “critical slowing down”(CSD). When the system approaches a bifurcation, its re-laxation time after a perturbation grows asymptoticallyand this divergence is referred to as CSD [11]. CSD isoften associated to an increase of the variance and of theautocorrelation of a variable of the system [12]. Nev-ertheless, it has been observed that these indicators arenot always reliable for alerting on an incoming bifurca-tion [13, 14].On the other hand, real systems evolve towards a bifur-cation because one or more parameters are changing in time. For example, the level of CO in the atmosphere isan evolving parameter that may lead the earth’s climatesystem to a bifurcation [15].In this paper we address the fundamental questionwhether CSD is always a good indicator of an incom-ing bifurcation in a system where a parameter is linearlychanging in time. By definition, CSD can be identifiedby perturbing the dynamical system. Unfortunately, inreal systems, this perturbation cannot be implemented inthe variables but rather in the parameters that are acces-sible in the experiments. Hence, a second question thatwe address here is whether a perturbation of an evolvingcontrol parameter can be a reliable probe for testing theoccurrence of a bifurcation. We answer to these questionsby presenting a real system with a time swept parameterwhere CSD appears only after the bifurcation has alreadyoccurred, hence when it might be too late to reverse thechange in behavior. Furthermore, we show that a per-turbation in the accessible control parameter is unableto provide any indication, nor on the occurrence of CSDnor on the bifurcation crossing.We begin by considering a simple two-dimensional dy-namical system describing a class-B laser [8], dS/dt = − S (1 − N ) ,dN/dt = − γ ( N − A + SN ) . (1)Here S is proportional to the light intensity and N tothe atomic population inversion; A is proportional to thepump and γ is the ratio between the decay rates of thepopulation inversion ( γ p ) and the intensity ( γ i ) . Thetime t is normalized to γ i . This dynamical system ex-hibits a transcritical bifurcation at A = 1. For A <
S, N ) = (0 , A ) is stable. For
A >
S, N ) = ( A − ,
1) is stable. We vary the pump param- a r X i v : . [ n li n . PS ] J u l FIG. 1. Intensity, S , and population inversion, N , as afunction of the pump, A . The pump is swept at a velocity b =0 . γ = 0 .
01. The initial conditions are S = 0 . , N =0 . , A = 0 .
8. The arrows indicate how S and N evolve asthe pump is increased and then decreased. The vertical arrowshows that N = 1 is reached when A = 1 . eter A with a triangular ramp of speed b , always smallerthan the decay rate of the variables of the system: A ( t ) = A + bt for t ≤ t A ( t ) = A + bt − b ( t − t ) for t ≤ t ≤ t (2)here A is the initial value of the pump and t is theduration of the ramp-up and of the ramp-down.The evolution of the laser intensity S as a functionof the pump parameter A is plotted in Fig. 1. S growssignificantly at a value well beyond the bifurcation point A = 1. This delayed reaction of a laser when the pump isswept across the threshold was studied theoretically [16]and experimentally [17, 18]. Critical slowing down wasput in evidence in [18] by measuring the asymptoticalgrowth of this delay as a function of the speed of thepump change. For a vanishing speed this delay diverges,thus revealing the presence of CSD at the laser threshold.As shown in Fig. Fig. 1, the intensity remains close tozero on a large interval during which the pump continuesto grow beyond the threshold value. Hence the systemaccumulates energy, which is suddenly released leadingto a spike-like variation of the intensity. If the system isunder damped ( γ < S in the laserintensity at different values of the pump parameter A .We measure the time taken by the perturbation to de-crease to 1/e of its initial value. Our results are plottedin Fig. 2. We notice that, for slow ramps, the relaxationtime diverges at the bifurcation point, i.e. when we ap-proach A = 1, as demonstrated in [18]. However, for FIG. 2. Relaxation time T of the laser intensity S afterthis variable was perturbed by a short pulse as a functionof the pump value A . Three values of the sweep velocity b are considered, while γ = 2 . × − is kept constant. Theoccurrence of CSD is marked by the asymptotic growth of T .For increasing b , CSD takes place at values of A larger thanthe bifurcation point ( A = 1, indicated with a dashed line). larger values of b , CSD does not take place at A = 1 butat a higher value of the pump parameter, which increaseswith b .In order to explain these observations, we note that,when S ≈ (cid:15) of the intensity is governed by: d(cid:15)dt = (cid:15) ( N −
1) (3)The relaxation time diverges when d(cid:15)/dt = 0, so when N becomes equal to 1. Importantly, the amplitude of theperturbation has no effect whatsoever during this stage( S ≈ t ≤ t ), the equationgoverning the evolution of N is: dNdt = − γ ( N − A − bt ) , (4)whose solution is N ( t ) = A + bt − b (1 − e − tγ ) /γ [21].Then, by imposing N = 1, we can calculate analyticallythe critical pump value ( A c ) at which the relaxation timediverges and CSD takes place: A c = 1 + bγ + bγ W (cid:104) − e − γb (1 − A ) − (cid:105) , (5)with W being the Lambert w function. Hence, for theparameters used in Figs. 1 and 2, A c depends mainly onthe ratio b/γ . As this ratio increases, A c grows abovethe value at which the bifurcation takes place ( A = 1).The effect of A on A c is negligible provided that A < − b/γ . In agreement with this analytical estimation, inthe simulations, using the parameters of Fig. 2, one finds A c = 1 . .
18 and 1 .
37 for b = 1 × − , 5 × − and1 × − respectively, while in Fig. 1, N = 1 is reachedwhen A = 1 .
05, indicated by the vertical arrow.Therefore, we have identified a system with a time-swept parameter in which CSD takes place well beyondthe bifurcation point, contradicting the common beliefthat CSD is an indicator of an upcoming bifurcation.Two ingredients are needed for the dynamical system tobehave in this counterintuitive manner: a fast sweepingrate of the parameter and low dissipation. The systemconsidered is a class-B laser where γ (the ratio betweenthe decay rates of the population inversion, γ p , and theintensity, γ i ) is significantly smaller than one.To meet this requirement we perform experiments witha diode-pumped solid state lasers (SSL) Nd:YVO4 emit-ting at 1.060 µ m. In this laser γ p is of the order of 2 × ,while γ i is 5 × , leading to γ ≈ × − (see Supple-mentary Material).SSL threshold is observed for a bias current J of thediode pump J = J th = 147 mA. The diode pump lasercan be modulated by a triangular ramp applied to its biascurrent, hence sweeping linearly the pump intensity froma zero level ( J = 88 mA, which corresponds to the diodepump threshold) up to 1.4 times the threshold value ofthe SSL ( J = 1 . × J th = 208 mA). The laser packageis thermally stabilized in a temperature range where theSSL emits on the same single longitudinal mode in thewhole swept pump range. The input and output signals(the bias current of the diode pump and the intensity ofthe SSL respectively) are monitored on a digital oscillo-scope. The ramp duration can be varied from 0.05 s to0.25 ms. The speed of the fastest ramp, as defined inEq. 2, is b = 1 . × − , hence b/γ = 0 .
28 (see Sup-plementary Material). According to Eq. (5), this uppervalue of b/γ , together with the possibility of controllingexperimentally b , makes this laser an ideal system to testthe prediction of this equation. Unfortunately, as in themajority of real systems, it is not possible to perturb di-rectly the laser variables ( S, N ) to probe the occurrenceof CSD. However, real systems can be perturbed throughtheir control parameters and we may check their influenceon the variables.The SSL laser intensity output versus the time-varyingpump level is shown in the upper panel of Fig. 3 fora modulation rate of 100 Hz. We notice, in agreementwith Fig. 1, that the laser intensity grows significantlyonly when the pump current is well above the thresh-old (the intensity spike occurs at J = J on =175.5 mA,which corresponds to 1 . J th ). The first lasing peak is fol-lowed by damped relaxation oscillations whose frequencyincreases as the pump increases. The delayed responseof the SSL in terms of the pump level has been investi-gated for different speed of the bias current ramp b andwe have observed that it follows the b − law predicted a b c defgha b c de f g h FIG. 3. Upper Panel: Laser output intensity as a functionof the bias current of the diode pump that is swept in timeby using a triangular ramp (see text for details). The in-tensity trace obtained during the positive (negative) slope ofthe ramp is displayed in blue (orange). Panels a-h): A shortpulse (1 µs width and 40 mA height) is superimposed overthe pump ramp at the positions indicated by the arrows onthe upper panel. The effect of each perturbation on the laserintensity is shown in the corresponding panels. in [16]. This delay is almost absent when ramping downthe pump current and the laser switches off at J ≈ J th .The difference between the pump value at which the laserstarts to emit ( J = J on ) and the pump value at which itswitches off ( J ≈ J th ) leads to the well-known dynamicalhysteresis of Fig. 3, which has also been observed in [18].For the low modulation rate used in Fig. 3 we estimate b = 5 . × − and b/γ = 0 .
014 (see Supplementary Ma-terial). Hence, according to Eq. (5), CSD is expected tooccur very close to the SSL threshold ( J = J th ). Wetest the occurrence of CSD by adding to the bias currentof the diode pump a perturbation pulse which is syn-chronous with the current ramp. By varying the phaseof the two signals we can place the perturbation at arbi-trary positions of the ramp and analyze the response inthe intensity variable. We apply a pulse of 40 mA with aduration of 0.5 µs at full-width half-maximum (FWHM).We superimpose it to the pump ramp and in Fig. 3, weshow the most relevant positions, marked by the arrowsin the upper panel.The evolution of the perturbation depends clearly onits position on the ramp, as shown by the panels a)to h). When the pump current is ramped up and thepulse is applied before the laser emits the first spike andswitches on (panel a), the intensity is not affected andit remains at the level of the experimental noise. Thisis observed for any position of the perturbation in theinterval 0 < J < J on = 1 . J th . If the perturbation isapplied after the laser has switched on, the pulse mayenhance the next relaxation oscillation peak (panel b).Instead, when it is applied between two relaxation oscil-lation peaks, it will decrease the amplitude of the nextrelaxation oscillation. In any case, the perturbation pulseinduces a new transient in the relaxation process startedafter the laser switch-on. This can be seen in panels c)and d) where the perturbation is applied when the laseroscillations are significantly damped. One can notice thatthe relaxation is faster when the perturbation is appliedcloser to the top of the ramp, i.e. at the maximum valueof the pump current. When the pump current is rampeddown, the evolution of the perturbation is not interact-ing with another relaxation process and therefore, it ismore clearly visualized: during the ramp down the per-turbation induces damped relaxation oscillations. As theperturbation is applied closer to the bifurcation pointwhere the laser switches off, the damping time increaseswhile the relaxation oscillations decreases (panels e,f,g).Finally, after the laser switches off, the perturbation doesnot induce any reaction on the intensity variable (h).These experimental evidences indicate that no signa-ture of CSD, nor of the bifurcation crossing at J = J th ,can be found by perturbing the pump parameter whenthe system evolves from the off-state to the on-state.This surprising behavior is observed for any speed ofthe ramp, even for the highest ones, where CSD is ex-pected to occur well-beyond the threshold value of theSSL. When b is increased the laser switches on at an in-creasing pump level (for example, with a ramp durationof 0.5 ms, J on = 1 . × J th ) according to the law pre-dicted in [16], and no effect on the intensity output isnoticed when the pump perturbation is applied in the in-terval 0 < J < J on . Instead, the perturbation pulse doeshave an effect on the intensity output when the laser is inthe on state. In this case, intensity exhibits a spike fol-lowed by damped relaxation oscillations whose frequencyand damping rate decrease as the perturbation is appliedcloser to the bifurcation point ( J = J th ).In order to understand why perturbing the control pa-rameter is not a reliable method to probe the occurrenceof CSD in our laser, we have used Eqs. 1 and 2 to ana-lyzed numerically the effect of a short pulse in the pumpparameter, i.e., A ( t ) = A ramp ( t ) + A p ( t ), where A ramp ( t )is the triangular signal described by Eq. (2) and A p ( t ) is ashort rectangular pulse. The results obtained, displayedin Fig. 4, are in very good agreement with the experimen-tal findings. We remark that for the parameters used inFig. 4 b/γ = 0 .
05, and therefore, according to Eq. (5)CSD occurs at A ∼ FIG. 4. Top panel: Intensity dynamics as a function of thepump. The parameters are as in Fig. 1. The arrows andvertical lines indicate the position at which we apply a shortpulse to the pump (see text for details). Central (bottom)panels: intensity dynamics when the pulse is applied duringthe upward (downward) ramp. noise show similar results (see Supplementary Material).The response of the system to a short perturbation inthe pump can be understood by analyzing the structureof the equations. When the laser is off, the intensity S vanishes and N must, in principle, follow the pump pa-rameter A . However, being N a slow variable ( γ << A , as, for ex-ample, when A is perturbed by a short pulse. Hence, thepump pulse does not affect the value of the variable N which will just continue to follow the pump ramp and S will remain close to zero, even if the perturbation pulseis applied when A > A c , i.e. when the pump parameteris beyond the critical point where N = 1 and CSD oc-curs. In fact, no response in the S variable to the pumppulse can be observed before the laser switches on. Afterthe first laser spike, S > N . In thiscondition, the relaxation process following the pump per-turbation is observable in the variable S , and an effect ofthe pump pulse on the intensity output can be measured;however, this occurs only after the laser has turned on.In conclusion, we have shown that, in a low-dissipationsystem with a control parameter that is swept linearly intime, CSD is not always a reliable indicator of an in-coming bifurcation. By considering a two-dimensionalreal system featuring a transcritical bifurcation, we havedemonstrated that CSD may occur beyond the bifur-cation point, which makes it useless for alerting of anincoming behavioral change of the system.Moreover, wehave shown that a perturbation of an evolving parame-ter might not able to identify CSD: this occurs when theparameter affects directly a slow variable. In this case, afast perturbation pulse may leave this variable unchangedand will have no effect on the system output. While theseresults can be generalized to any dynamical system hav-ing a dimension ≥ ∗ Corresponding author: [email protected][1] M. Scheffer, J. Bascompte, W. A. Brock, V. Brovkin,S. R. Carpenter, V. Dakos, H. Held, E. H. Van Nes,M. Rietkerk and G. Sugihara, Nature , 53 (2009).[2] M. Scheffer, S. R. Carpenter, T. M. Lenton, J. Bas-compte, W. Brock, V. Dakos, J. Van De Koppel, I. A.Van De Leemput, S. A. Levin, E. H. Van Nes, M. Pascualand J. Vandermeer, Science , 344 (2012).[3] N. Malik, N. Marwan, Y. Zou, P. J. Mucha andJ. Kurths, Physical Review E - Statistical, Nonlin-ear and Soft Matter Physics (2014), 10.1103/Phys-RevE.89.062908.[4] B. Litt, R. Esteller, J. Echauz, M. D’Alessandro, R. Shor,T. Henry, P. Pennell, C. Epstein, R. Bakay, M. Dichterand G. Vachtsevanos, Neuron , 51 (2001).[5] P. E. McSharry, L. A. Smith, L. Tarassenko, J. Mar-tinerie, M. Le Van Quyen, M. Baulac and B. Renault,Nature Medicine , 241 (2003).[6] T. M. Lenton, H. Held, E. Kriegler, J. W. Hall, W. Lucht, S. Rahmstorf and H. J. Schellnhuber, Proceedings of theNational Academy of Sciences of the United States ofAmerica , 1786 (2008).[7] N. M. Granese, A. Lacapmesure, M. B. Ag¨uero, M. G.Kovalsky, A. A. Hnilo and J. R. Tredicce, Optics Letters , 3010 (2016).[8] C. Metayer, A. Serres, E. J. Rosero, W. A. S. Barbosa,F. M. De Aguiar, J. R. Rios Leite and J. R. b. Tredicce,Optics Express , 19850 (2014).[9] J. N. Thompson, Trends in Ecology and Evolution ,327 (1998).[10] V. Dakos, S. R. Carpenter, W. A. Brock, A. M. Ellison,V. Guttal, A. R. Ives, S. K´efi, V. Livina, D. A. Seekell,E. H. van Nes and M. Scheffer, PLoS ONE (2012),10.1371/journal.pone.0041010.[11] H. Mori, Prog. Theo. Phys. 30, 576 (1963).[12] V. Dakos, E. H. van Nes, P. D’Odorico, M. Scheffer, Ecol-ogy 93, 264 (2012).[13] S. J. Burthe, P. A. Henrys, E. B. Mackay, B. M. Spears,R. Campbell, L. Carvalho, B. Dudley, I. D. M. Gunn,D. G. Johns, S. C. Maberly, L. May, M. A. Newell,S. Wanless, I. J. Winfield, S. J. Thackeray, F. Dauntand C. Allen, Journal of Applied Ecology , 666 (2016).[14] V. Guttal, S. Raghavendra, N. Goel and Q. d. Hoarau,PLoS ONE (2016), 10.1371/journal.pone.0144198.[15] T. M. Lenton, V. N. Livina, V. Dakos, E. H. van Nes, M.Scheffer, Phil. Trans. Royal Soc. A 370, 1185 (2012).[16] P. Mandel and T. Erneux, Physical Review Letters ,1818 (1984).[17] W. Scharpf, M. Squicciarini, D. Bromley, C. Green, J. R.Tredicce and L. M. Narducci, Optics Communications , 344 (1987).[18] J. R. Tredicce, G. L. Lippi, P. Mandel, B. Charasse,A. Chevalier and B. Picqu´e, American Journal of Physics , 799 (2004).[19] F. T. Arecchi, G. L. Lippi, G. P. Puccioni and J. R.Tredicce, Optics Communications , 308 (1984).[20] D. Bimberg, K. Ketterer, E. H. Bottcher and E. Scholl,Int. J. Electron. 60, 23 (1986).[21] If the pump ramp starts well below the threshold, theinitial condition of the population inversion is N (0) = A0